Saclay-t17/081

CERN-TH-2017-119

SISSA 30/2017/FISI

The Other Fermion Compositeness

Brando Bellazzini a, b, Francesco Riva c, Javi Serra c and Francesco Sgarlata d

a Institut de Physique Theorique, Universite Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, Franceb Dipartimento di Fisica e Astronomia, Universita di Padova, Via Marzolo 8, I-35131 Padova, Italy

c CERN, Theory Department, CH-1211 Geneve 23, Switzerlandd SISSA International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy

Abstract

We discuss the only two viable realizations of fermion compositeness described

by a calculable relativistic effective field theory consistent with unitarity, crossing

symmetry and analyticity: chiral-compositeness vs goldstino-compositeness. We

construct the effective theory of N Goldstini and show how the Standard Model

can emerge from this dynamics. We present new bounds on either type of com-

positeness, for quarks and leptons, using dilepton searches at LEP, dijets at the

LHC, as well as low-energy observables and precision measurements. Remark-

ably, a scale of compositeness for Goldstino-like electrons in the 2 TeV range is

compatible with present data, and so are Goldstino-like first generation quarks

with a compositeness scale in the 10 TeV range. Moreover, assuming maximal

R-symmetry, goldstino-compositeness of both right- and left-handed quarks pre-

dicts exotic spin-1/2 colored sextet particles that are potentially within the reach

of the LHC.

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1 Motivation

The LHC defines the frontiers of our exploration of the universe at microscopic scales. Its

primary focus so far has been the search for new physics in the form of narrow resonances,

often associated with weakly coupled (calculable) physics beyond the Standard Model (SM).

The incredible amount of data that is and will be accumulated at the LHC gives access to

unprecedented accuracy in our knowledge of SM processes. A crucial question is then to

understand what kind of physics can be tested with all this new information. Amusingly,

strongly (as opposed to weakly) coupled dynamics, associated with the compositeness of SM

particles, can produce sizable deviations in their interactions at high energy and become the

principal target of these SM precision tests [1–10].

In this article we present a perspective on how the SM can emerge as the result of such

a strong dynamics, a perspective on how high-energy compositeness can appear to a low-

energy observer. We are interested in situations where the Compton wavelength of a particle

∼ m−1 is much larger than the microscopic scale ∼ m−1∗ of the confining theory, so that

it effectively appears elementary at energies of order its mass.1 A hierarchy between these

different scales can be naturally generated by the presence of certain approximate symmetries

that characterize the strong dynamics but are explicitly broken by m and by the other SM

interactions. Notable examples are non-linearly realized symmetries that protect the mass

of Nambu-Goldstone modes, or chiral symmetry that protects that of fermions. In practice,

the imprint of compositeness can be captured in an effective Lagrangian by irrelevant but

strong operators of dimension D > 4, weakly deformed by marginal or relevant (SM-like)

interactions with D ≤ 4. At low energy only the latter survive and particles look SM-like,

while at high-energy the former grow and the new strong interaction is gradually revealed.

The more irrelevant the new interactions, the more this scenario would resemble the SM

at low energy, simply because the effect of compositeness would vanish more rapidly as the

energy that is used to probe it decreases. It is therefore crucial to understand how well

can a framework of compositeness in the multi-TeV region (accessible at present or near-

future colliders) fake the SM: how irrelevant can these new interactions be? From a low-

energy perspective one would see no obstruction in arbitrarily soft interactions, but dispersion

relations can be used to show that the basic requirement of unitarity of the microscopic (UV)

theory, imposes certain positivity constraints that guarantee the existence of unsuppressed

dimension-8 operators [12–14]. Given the SM field content and limiting our discussion to

lepton and baryon number preserving effects, this implies that the leading BSM effects must

be characterized either by operators of dimension-6 or by dimension-8, but not higher. While

most of the literature focuses on the former case, in this article we discuss instead the latter

situation in which the leading BSM interactions saturate the unitarity bound and arise at

1Note therefore that we are not concerned here with compositeness of the type the proton exhibits for

instance. This would give rise to large non-standard effects for the SM fermions already at energies comparable

to their mass, effects which have been long ruled out experimentally. Instead our focus is on the class of

compositeness that can only be efficiently probed at higher energies, in the spirit of Ref. [11].

2

dimension-8: these are theories that flow maximally fast to the SM as energy is decreased.

In fact, systems with very soft behavior have been studied in the context of scalars includ-

ing the Higgs [7,15], and vectors [7]. In this article we focus on fermions, and go beyond the

traditional paradigm where fermion compositeness relies on chiral symmetry and the asso-

ciated four-fermion dimension-6 operators. We consider fermions whose composite nature is

dominantly captured by dimension-8 operators (four-fermion with two extra derivatives) via a

non-linearly realized supersymmetry that forbids operators with D < 8. That is, (some of) the

SM fermions are pseudo-Goldstini, in a modern generalization of the Akulov-Volkov theory

for the neutrino [16,17]. For this reason we dub this power-counting goldstino-compositeness,

as opposed to the well-known chiral-compositeness based on chiral symmetry.

Present and future colliders have incredible sensitivity to modifications of the properties

of light fermions. We find this a strong motivation to understand fermion compositeness

from both a theoretical and phenomenological point of view and to assess in which ways

processes involving fermions can carry information about any structurally robust BSM dy-

namics, even if exotic. This ambitious task is simplified by our observation that there exist

only two patterns of fermion compositeness compatible with prime principles. The traditional

chiral-compositeness of fermions, which gives rise to dimension-6 operators, is incredibly well

constrained by LHC data for the light quarks, leaving no hope of ever testing it directly on an

earth-based collider (see section 5 or Ref. [3]). On the other hand, non-linearly realized su-

persymmetry, i.e. goldstino-compositeness, provides an alternative explanation for the small

fermion masses, while allowing large effects in the more irrelevant dimension-8 operators,

whose collider constraints, as we will show here, are much less severe. Interestingly, the fast

decoupling of the Goldstino interactions becomes distinctive as well when considering diboson

production, whose leading modifications come from dimension-8 operators when the Higgs or

the transverse gauge bosons are also composite.

This paper is organised as follows. In section 2 we present our picture for goldstino-

compositeness based on approximate extended supersymmetries. We construct the effective

theory of Goldstini in section 3. In this theoretically oriented section we discuss the geom-

etry of the coset space associated to supersymmetry breaking, the general interactions of

the Goldsitini, and the embedding of quarks and leptons. In section 4 we understand the

deformations induced by the explicit breaking of supersymmetry. The phenomenology of our

scenario is covered in sections 5 and 6, where we discuss, respectively, the 2 → 2 scattering

process that best probe the compositeness of quarks and leptons and the exotic “quixes” that

are predicted by maximal R-symmetry.

2 Pseudo-Goldstini

We consider a strongly interacting supersymmetric sector. It confines breaking SUSY spon-

taneously and leaving only massless Goldstini in the infrared (IR) spectrum, that we identify

with (some of) the SM fermions; in addition to kinetic terms, Goldstini have self-interactions

3

that start at dimension-8. This picture is deformed by non-supersymmetric SM couplings,

which we take to be small compared with the new sector’s coupling, thus we treat them as

perturbative deformations of the exact Goldstino limit.2 Yet, these small effects are marginal

and at small enough energy they dominate, while at high energy Goldstino self-interactions

become more important.

We will study scenarios where either one, several, or even all of the quarks and leptons are

(pseudo-)Goldstini, implying the existence of 1 ≤ N ≤ 84 supersymmetries (see section 3.3)

– numbers at which the reader might be willing to raise an eyebrow. It is indeed well known

that complete massless supermultiplets (i.e. within a linearly realized SUSY with massless

particles) and N > 8 supercharges imply the existence of massless higher-spin states which

are pathological, in flat space, on very general grounds [19–21] (see e.g. [22] for a review).

Yet, the roots of these arguments are based on the IR properties of the states, i.e. the soft

scattering limits of the S-matrix elements or the existence of global charges, which is exactly

the regime where one is sensitive to the spontaneous symmetry breaking effects, i.e. the soft

masses. A spontaneously broken extended SUSY does not predict those massless higher spins

at all.3 In non-linearly realized SUSY, the would-be higher-spin superpartners are actually

multi-particle states obtained by including Goldstino insertions that do in fact raise/lower

the spin, but without producing single particle states.4

One can take a step further and argue that the very existence of N > 8 implies the only

consistent phase of SUSY in flat space is the broken one, which in turn requires the existence

of fermions much lighter than the cutoff, as we observe in nature. There could then be

a fascinating link between the (nearly) vanishing cosmological constant and the existence of

(light) spin-1/2 fermions in the spectrum. Moreover, it could be possible that a UV completion

in the unbroken phase needs to be formulated in AdS space, given the need to inject a positive

contribution to the vacuum energy to move to the broken phase. Notice that massless higher

spins in AdS pose no problem [22,25,26].

There is in fact another difficulty with N ≥ 8: the impossibility of interactions of renor-

malizable dimension D ≤ 4, in a Lagrangian description. The only interactions that can be

written compatibly with these extended supersymmetries are highly irrelevant. This suggests

that, if a UV completion exists, is not of a weakly coupled kind and does not necessarily rely

on a Lagrangian or effective description. It may well be that a UV completion expressed as

local Lagrangian is made of a series of higher-dimension operators that become important at

energies of order the higher-spin states mass, yet resulting in a well-defined S-matrix at any

finite energy. This is somewhat analogous to what happens in Vasiliev’s theories in curved

AdS space [25,26], where the cutoff is given by the cosmological constant itself, i.e. the deep-

est IR observable, such that theory is never weakly coupled in the UV, where the curvature

2Another interesting application of this ideology is Goldstino Dark Matter [18].3Incidentally, in our framework neither the gauge bosons nor the graviton are actually part of the strong

sector (more on this below), evading even more the existence of massless supermultiplets.4This is made manifest for instance in the constrained superfield formalism [23,24], where the scalar partner

of the Goldstino is a pair of Goldstini, X = χ2 + θχ+ θ2F .

4

could in principle be neglected. Another example is the high-energy limit of string theory [27],

where infinitely many higher-spin conserved currents appear to be restored, corresponding to

the tensionless limit.

Alternatively, extended SUSY could just be realized as an emergent symmetry which

(re)appears in the IR from UV dynamics that do not necessarily exhibits such a symme-

try [28–30]. Yet another option is to consider several almost sequestered N ≤ 8 supersymmet-

ric sectors which are linked by an approximate permutation symmetry, e.g. ZMo [N -SUSY]M

(where M is a flavor index).

In any case, since there are no manifest obstructions to the presence of several super-

charges, we assume that a strongly coupled sector that breaks an extended SUSY indeed

exists. We call g∗ . 4π its strong coupling at the scale of confinement m∗. We will consider

first the limit of rigid supersymmetry and no gravity (MPl → ∞), as well as vanishing SM

couplings gSM ≡ {g, g′, gs, Yψ} → 0. In this limit we assume SUSY is spontaneously broken

with order parameter F ∼ m2∗/g∗, resulting in N massless Goldstini.

A finite MPl corresponds to the gauging of Poincare symmetry, which in the spontaneously

broken supersymmetric context implies supergravity with massive Gravitini [31]. This is not

how nature looks like: the SM fermions have spin 1/2, excluding N supergravities. Instead,

finite MPl with rigid supersymmetry introduces explicit SUSY breaking (the graviton has no

superpartner) that can potentially propagate to the matter sector, giving large contributions

to relevant operators (e.g. ∼M2Pl contributions to scalar masses) and masses to the Goldstini.

However, in theories without elementary scalars [28], or strongly coupled theories where the

|φ|2 operators have D & 4 [29,30], SUSY is still approximately preserved. Moreover, an exact

N = 1 supergravity in addition to N rigid supersymmetries would be enough to protect scalar

masses and preserve the necessary amount of SUSY [32].

The strongly interacting supersymmetric sector is also coupled to an elementary sector

which includes, possibly, but not necessarily, elementary (non-supersymmetric) scalars, gauge

bosons and some fermions. The elementary sector then breaks SUSY explicitly, but in a way

that can be kept under control, given that gSM � g∗ is a small perturbation of the undeformed

theory, in the spirit of Ref. [33]. In particular, the SM gauge vectors couple to (part of) the

conserved currents in the supersymmetric sector, which are associated to an unbroken R-

symmetry under which the Goldstini-like SM fermions transform.5 We further assume that

R-symmetry includes (part of) the SM flavor group, broken only by the Yukawa couplings to

the Higgs boson. This construction reproduces a form of MFV [36] that successfully surpasses

flavor constraints even for a low SUSY-breaking scale.

5There is a priori a different starting point to this construction in which R-symmetry is gauged (necessarily

together with supergravity) [34]. However, in such theories R-symmetry is broken at very high scales by the

VEV of the superpotential, in order to cancel the cosmological constant; see however [35] for a possible caveat.

5

3 The Effective Goldstini Theory

Following the outline in the previous section, we first construct the Goldstini EFT, in the

limit where (rigid) explicit SUSY breaking effects vanish, that is gSM → 0 and MPl → ∞.

This section is rather formal: the results relevant for the rest of the paper are summarized in

Tables 1, 2 and 3.

3.1 The geometry of N Goldstini

As for every Goldstone field, the Goldstini χi can be chosen to parametrize the coset space as-

sociated to spontaneous SUSY breaking [16,38–41].6 They provide a map from the spacetime

to an element of the SUSY transformations, x→ g(x), up to the identification g(x) ∼ g(x)h(x)

where h(x) is an element of the unbroken symmetry group that includes (homogeneous)

Lorentz transformations.7 At the Lagrangian level, this identification is realized as usual by

constructing a gauge theory invariant under local Lorentz transformations, a gravity theory

of a sort, although with a non-dynamical composite vielbein E aµ (χ) made of Goldstini. To

this end, we choose the following coset representative

U(x, χ(x)) ≡ ei(χ(x)Q+χ†(x)Q†)eixµPµ , (1)

where χQ and χ†Q† are shorthand for χαi Qiα and χ†iαQ

† αi respectively, where the sum over

repeated indexes is always understood unless stated otherwise. In the following, both spinor

and flavor contractions are understood. The relevant part of the extended SUSY algebra we

consider is

{Qiα, Q

†β j} = 2σµ

αβPµδ

ij , [Pµ, Q

iα] = [Pµ, Q

† iα ] = {Qi

α, Qjβ} = {Q† iα , Q

† jβ} = 0 , (2)

where the latin indexes i, j = 1, . . . ,N label the supercharges, while the greek indexes are

spinorial in the (1/2, 0) (undotted) or (0, 1/2) (dotted) representation of the Lorentz group

SO(1, 3) ∼ SU(2) × SU(2). We assume no central charge since, as explained above, we are

eventually interested in identifying the unbroken R-symmetry GR ⊆ U(N ) ∼ SU(N )R ×U(1)R, or some of its subgroups, with the SM flavor and gauge groups.

Under a global transformation g, the representative element U is moved into another group

element of the form gU = U ′h which defines the transformation

U(x, χ(x))→ U ′(x′, χ′(x′)) = gU(x, χ(x))h−1 , (3)

for the coset representative, hence for the Goldstini.

6Different and yet equivalent parametrizations are often adopted for the Goldstini, like the one provided

by the constrained superfield formalism, see e.g. [24, 37] and references therein. Constrained superfields are

convenient whenever the Goldstini couple to IR fields that fill complete SUSY multiplets, i.e. in linear SUSY

representations. This is never the case in our EFT.7Spacetime translations are effectively considered as broken generators since they are non-linearly realized

on spacetime, which is nothing but the coset Poincare/Lorentz, x→ x+ c (see e.g. [42]).

6

The Goldstini transform linearly under the unbroken symmetries. In particular, they

transform under Lorentz as ordinary two-component spinors.8 Similarly, for an unbroken

R-symmetry GR the Goldstini χ and χ† carry a linear representation, e.g. the fundamental

representation of maximal R-symmetry

[RaSU(N )R

, Qi] = (T a)ij Qj , [RU(1)R , Q

i] = Qi ,

[RaSU(N )R

, Q†i ] = −(Ta)i

jQ†j , [RU(1)R , Q

†i ] = −Q†i , (4)

where T a are the N 2 − 1 traceless and hermitian N –by–N generators of SU(N )R.

Under spacetime translations T = eiaµPµ , U ′ = TU where x′ = x + a and χ′(x′) =

χ(x(x′)) = χ(x′ − a) as usual.

Under the action of the broken SUSY generators gξ = exp[iξjQj + iξj†Q†j], we get instead

a non-linear realization:

gξ U(x, χ(x)) = U ′(x′, χ′(x′)) →{χ′(x′) = χ(x(x′)) + ξ

x′µ(x) = xµ − iχ†(x)σµξ + iξ†σµχ(x). (5)

Indeed, under an infinitesimal ξi it corresponds to the following non-linear action of the SUSY

transformations

χ(x)→ χ′(x) = χ(x′(x)) + ξ = χ(x) + ξ − vµ(ξ, χ)∂µχ(x) + . . . (6)

where vµ(ξ, χ(x)) ≡ −i(χ†(x)σµξ − ξ†σµχ(x)

).

For matter fields (scalars and spinors) the non-linear representation is the same as for the

Goldstini except that it does not include the Grassmannian shift ξ, namely

Φ(x)→ Φ′(x) = Φ(x′(x)) = Φ(x)− vµ(ξ, χ(x))∂µΦ(x) + . . . (7)

3.1.1 Covariant derivatives and Maurer-Cartan form

Notice that derivatives transform non-covariantly, that is differently than the undifferentiated

fields.9 In order to deal with proper covariant transformations of derivatives, it is useful to

introduce the Maurer-Cartan 1-form in the SUSY algebra

(U−1dU)(x) = idxµE aµ

(Pa +∇aχQ+∇aχ

†Q†), (8)

where for future convenience we have factored out the coefficient E aµ of the momentum

E aµ = δaµ − iχj†σa∂µχj + i∂µχ

j†σaχj , (9)

∇aχj = (E−1) µa ∂µχj ≡ E µ

a ∂µχj , (10)

8Explicitly, we have LU(x, χ(x)) = Lei(χ(x)Q+χ†(x)Q†)L−1(Leix

µPµL−1)L = U ′(x′, χ′(x))L where one

can use LPµL−1 = PνΛνµ, LQiαL

−1 = Λ βα Qiβ , and LQ† αi L−1 = Λα

βQβi to get the Lorentz action on the

Goldstini, namely χ′α(x′) = χβ(x(x′)) Λ αβ with Λ α

β the spinorial (1/2, 0) representation of L.9Explicitly, ∂µχ(x) → ∂µχ(x′(x)) = (∂νχ)(x′(x)) ∂x

′ν

∂xµ = ∂µχ(x′(x)) − ∂νχ(x′(x)) vν(ξ, ∂µχ(x)) + . . . and

only the first term on the right-hand side corresponds to the covariant SUSY transformation.

7

with ∇aχj† = (∇aχ

j)†, and defined the inverse of E aµ as

(E−1) µa ≡ E µ

a , E µa E

bµ = δba , E a

µ Eνa = δνµ . (11)

Since the Maurer-Cartan form is invariant, (U−1dU)′(x′) = (U−1dU)(x), its change at a given

point is just the contraction with the Jacobian matrix. This tells us that the E aµ and E µ

a

transform as vielbeins

E aµ (x)→ E ′ aµ (x) =

∂x′ν

∂xµE aν (x′(x)) , E µ

a (x)→ E ′ µa (x) =∂xµ

∂x′νE νa (x′(x)) . (12)

It is therefore clear that ∇aχ transforms covariantly, as in Eq. (7),

∇aχ(x)→ (∇aχ)′(x) = (∇aχ)(x′(x)) . (13)

Gauge fields behave just as derivatives. The resulting gauge- and SUSY-covariant deriva-

tives read Da ≡ ∇a − iAa = E µa (∂µ − iAµ). Similarly, the covariant field strength is

Fab = E µa E

νb (∂µAν − ∂νAµ − [Aµ, Aν ]) . (14)

In our EFT we gauge an R-symmetry, as opposed to an ordinary global symmetry, which

forces us to break SUSY explicitly. However the gauge fields themselves could in principle

be part of the strong sector, that is composite [7]. In this case, one should dress with the

vielbeins only the ∂[µAν] part of the field strength and not its non-abelian part [Aµ, Aν ]. For

elementary gauge fields neither term should be dressed with Goldstini.

3.1.2 Effective metric and invariant measure

From the composite vielbeins one can define a composite effective metric

gµν(x) ≡ E aµ (x)E b

ν (x)ηab → g′µν(x) =∂x′ρ

∂xµ∂x′σ

∂xµgρσ(x′(x)) (15)

and use it to build the various invariants that do not involve fermions; the latter requiring

the vielbein too. The inverse metric is promptly found to be

gµν = E µa E

νb η

ab = ηµν +(iχ†σµ∂νχ+ iχ†σν∂µχ+ h.c.

)+ . . . (16)

Note that E aµ ηabg

µν = E νb , meaning that we can lower and raise the various indexes with

the appropriate metrics. The determinant of the Vielbein

detE ≡ detE aµ (x) =

√− det gµν (17)

yields the SUSY-invariant measure

d4x detE(x)→ d4x|∂x′

∂x| detE(x′(x)) = d4x′ detE(x′) . (18)

8

The coset-ology allows us to work with objects that transform covariantly even with respect

to the broken generators, upon vielbein and metric contractions. These objects admit an

interesting geometrical interpretation associated to the extended superspace (xa, θαi , θ† iα ) where

the 1-forms ωa = dxa−iθ†σadθ+idθ†σaθ, dθ and dθ† are invariant under SUSY transformations

δξ(xa, θ, θ†) = (−iθ†σaξ + iξ†σaθ, ξ, ξ†) [16]. These vector 1-forms are pulled-back to E a

µ dxµ,

∂µθdxµ and ∂µθ

†dxµ via the map xµ → (xa(x), θα i(x), θ†α i(x)). These forms are nothing

but the coordinates with respect to the basis Pa, Q, Q† of the Maurer-Cartan form, up to

identifying the Goldstino θ(x) = χ(x) as a space-filling brane in the extended superspace. The

invariant measure (18) is nothing but the induced volume element of the Goldstino brane.

Every field of the strong sector that interacts with the induced metric unavoidably interacts

with the Goldstini, resulting in model-independent effects. In addition, our construction allows

higher-derivative terms that are model-dependent, following from the direct interactions with

∇χ. We explicitly discuss these interactions in the next section. Among the model-dependent

contributions, new higher-derivative geometrical objects may appear as well, such as

F abc (x) ≡ E µ

b (x)E νc (x)

(∂µE

aν (x)− ∂νE a

µ (x))

= 2i∂bχj†σa∂cχj+2i∂cχ

j†σa∂bχj+. . . , (19)

which transforms covariantly, F abc (x) → F ′ a

bc (x′(x)) (thanks to the anti-symmetry µ ↔ ν)

and represents a sort of connection [∇a,∇b]Φ(x) = −F cab ∇cΦ(x).

3.2 The Goldstini Effective Action

We now have all the covariant ingredients to write an effective action,

SSUSY[χ,Φ] =

∫d4x detE L(∇aχ(x),Φ(x),∇aΦ(x), F a

bc (x), . . .) =

∫d4x

√− det g L (20)

which is manifestly invariant under SUSY transformations Eq. (6) should L be Lorentz sym-

metric.10 One can also consider nearly secluded sectors, each enjoying its own extended SUSY

simply by summing over actions of the type Eq. (20); see e.g. [43] for the case of N copies of

N = 1 SUSY. In the following we focus on a single sector with N supercharges in order to

avoid the proliferation of unknown constants of possibly disparate scales.11

3.2.1 Goldstini self-interactions: Akulov-Volkov

The most important SUSY-preserving Goldstini interactions are those in Eq. (20) with the

least number of derivatives. There is in fact a unique operator that yields both the leading

10In practice this requires that all the Lorentz indexes a, b, c,. . . must be saturated contracting them in pairs

with the Minkowski metric ηab, its inverse ηab, the fully anti-symmetric εabcd symbol, or the sigma matrices σa.

Likewise for the spinorial indexes. To construct an action that is also R-symmetric, the flavor indexes i, j, . . .

should be contracted among themselves with the relevant invariant tensors, such as δji . Notice that detE

alone is automatically, i.e. accidentally, invariant under maximal R-symmetry U(N )R ∼ SU(N )R × U(1)R.11However, one could in principle consider other global symmetries to reduce the unknowns. A trivial

example would be semi-direct products of the type ZM o [N -SUSY]M .

9

interactions and the kinetic terms. Not surprisingly, it is the most relevant operator in a

gravitational theory, the vacuum energy, injected at the SUSY phase transition, namely

LCC = −F 2 , (21)

which, dressed with the Goldstini, provides the extended-SUSY generalization [17] of the

Akulov-Volkov action [16]

S(0)SUSY =

∫d4x − F 2

√− det g = −F 2

∫d4x det

[δaµ − iχj†σa∂µχj + i∂µχ

j†σaχj]. (22)

The scale F is the SUSY-breaking scale. The action S(0)SUSY is accidentally invariant under

maximal R-symmetry U(N )R. This will be important when discussing flavor violations be-

yond those of the SM. Expanding the Vielbein’s determinant

detE = 1−(iχj†σµ∂µχj + h.c.

)(23)

+1

2

[(iχj†σµ∂µχj + h.c.

)2 − (iχj†σa∂µχj + h.c.) (iχj†σµ∂aχj + h.c.

)]+ . . .

we extract the kinetic term as well as the leading order interactions. The canonically normal-

ized Goldstini χ are√

2Fχ = χ. In the following we omit the tilde ˜, whenever clear. The

leading four-fermion interactions come from a dimension-8 operator that can be written as

−∫d4x

1

2F 2

(χ†jσ

a∂µχj)(

χ†i σµ∂aχ

i)

=

∫d4x

1

F 2(χ†i∂µχ

†j)(∂

µχiχj) (24)

after using the free equations of motion, integrating by parts, and using the Fierz indentity

∂νχ†βσµ ββσν αα∂µχα = −2∂µχ

† α∂µχβ. The Akulov-Volkov Lagrangian, with a single flavor

N = 1, can be recast as −χ† 2�χ2/(4F 2) upon integration by parts. This is the only operator

with four fermions and two derivatives that can be built with one fermion. With more flavors,

we note that any dimension-8 operator of this kind that is consistent with the absence of more

relevant interactions is necessarily selected by shift and spacetime transformations of the form

Eq. (6), which strongly suggests that they can be linked to a SUSY-breaking pattern, e.g. for

two flavors, N = 2 or (N = 1)2.

3.2.2 Model independent couplings to composite fields

Let us consider now the case where, in addition to the Goldstini, other particles are composite,

that is they emerge from the same sector that breaks SUSY spontaneously. These fields have

kinetic terms, provided by the SUSY-breaking sector itself, which need to be compensated by

Goldstini insertions to make up for the absent kinetic terms of the would-be superpartners.

Since the kinetic terms can always be rescaled to a canonical form, the resulting Goldstini

interactions are model independent, controlled by no free parameter except for the SUSY-

breaking scale F (as the Goldstini self-interactions).12

12In fact, this model independency is somewhat of a misnomer: the overall coefficient is controlled by F

alone because we assumed full compositeness. Should some of these non-Goldstini states X = ψ, Aµ or φ

10

composites naked terms SUSY dressing leading four-body interactions

χ −F 2 −F 2√− det g 1

F 2 (χ†i∂µχ†j)(∂

µχiχj)

χ, ψ i2ψ†i σ

µ∂µψi(x) + h.c. detE

(i2ψ†i σ

a∇aψi(x) + h.c.

)− 1F 2 (ψ†i σ

a∂µψi)(χ†jσ

µ∂aχj)

χ, F −14FAµνF

Aµν −√− det g 1

4FAµνF

Aρσg

µρgνσ − 14F 2F

AµνF

Aµρ

(iχ†i σ

{ρ∂ν}χi + h.c.)

χ, φ ∂µφi†∂µφi

√− det g gµν∂µφ

i†∂νφi1

2F 2

(iχ†jσ

{µ∂ν}χj + h.c.)∂µφ

i†∂νφi

Table 1: Model independent couplings for composite particles X = ψ,Aµ, φ. Partially composite particles

have these couplings weighted by their degree of compositeness squared |εX |2. If the scalar φ is a composite

NGB one should dress the whole kinetic term built with the D-symbol Dµi = ∂µφi +O(φ3).

Calling X = ψ, Aµ or φ any composite fermion, gauge boson or scalar, we construct their

SUSY-invariant kinetic terms by making the replacements ∂µX → ∇aX, d4x→ d4x√− det g,

ηµν → gµν , Aµ → Aa, Fµν → Fab. . . , from which we derive the so-called model independent

couplings of the composite states to the Goldstini. They are reported in Table 1, where we

have canonically normalized the Goldstini in the last column, used the free equations of motion

and integrated by parts. The curly brackets mean symmetrization, e.g. σ{µ∂ν} ≡ σµ∂ν+ σν∂µ.

Notice that all these interactions respect accidentally U(N )R.

When considering the couplings of the strong sector to external (gauge) sources, an im-

portant role is played in our construction by the R-symmetry current, the conserved Noether

current associated with infinitesimal R-symmetry transformations χ → χ − iωATAχ (TA

are the appropriate R-generators). This current can be found by noticing that in our ge-

ometrical construction Goldstini enter the action only through geometrical objects, so that

the R-current is directly related to the energy-momentum tensor (associated to the model-

independent universal contributions)

T µa = −δS/δE a

µ = −LE µa +detE

[(i

2ψi†σbE µ

b ∇aψi +∇aφi†∂νφig

µν + h.c.

)− FabFcbE µ

c

].

(25)

From this we find, for the R-current [44],

RAµ =1

F 2T µa χ†σaTAχ =

(χ†σaTAχ

)(δµa +

i

2F 2χj†σµ

←→∂ aχj + . . .

). (26)

Analogously, the SUSY currents can also be expressed in terms of the energy-momentum ten-

sor, namely Sµα j =√

2T µa χj†

β(σa)βα/F and S†µ αj =

√2T µ

a (σa)αβχjβ/F , with j = 1, . . . ,N .

3.2.3 Model dependent couplings to composite fields

The model dependent couplings may or may not be there, depending on the details of the UV

theory. Generically they do not respect maximal R-symmetry, but just some of its subgroups.

be partially composite instead, the would-be model independent couplings would actually get rescaled by the

degree of compositeness (squared), |εX |2, which measures how large a fraction of the kinetic term arises from

the SUSY-breaking sector. Note that εX 6= 1 necessarily breaks SUSY.

11

GR SUSY Lagrangian leading interactions

ψ = 1 cji detE (∇aχi†σb∇aχj)(ψ

†σbψ) cji1F 2 (∂νχ

i†σµ∂νχj)(ψ†σµψ)

π = 1, NGB dji detE(∇aχ

i†σb∇aχj)∇bπ five-body or ∝ mπ,mχ

π = , NGB c detE(∇aχ

†σbTA∇aχ) (iπ†TA

←→∇ bπ

)c 1F 2

(∂µχ

†σνTA∂µχ) (iπ†TA

←→∂ νπ

)Table 2: Model dependent couplings. For composite particles cji , d

ji , c = O(1).

In Table 2 we show a few illustrative examples assuming that the χi carry at least a U(1)R,

i.e. χi → χieiαR , that the scalar π is naturally light because it is a Nambu-Goldstone boson

(NGB), and we restrict to a singlet fermion. In the last row we kept the leading term in the

number of NGBs of the E-symbol, EAb = −iπ†TA←→∇ bπ + O(π4). We also enforced maximal

R-symmetry, GR = U(N )R, for simplicity. This can also be done for the first and second rows

by choosing cji , dji ∝ δji . Alternatively, choosing e.g. cji , d

ji = diag(c(N)IN×N , c(M)IM×M) with

N = N +M corresponds to realize linearly only the SU(N)R × SU(M)R × U(1)R subgroup

of U(N )R under which χ = (N, 1)x⊕ (1,M)−xN/M . The size of these interactions depends on

the assumptions about the UV theory, e.g. the size of couplings with additional heavy states.

The generalization to non-NGB scalars, non-singlet fermions, or other choices of conserved

R-symmetries and its representations is straightforward. For example, a scalar without a shift

symmetry has a model independent coupling proportional to its mass squared, −m2φ|φ|2 detE,

which contributes to a 6-body scattering.13 Higher-derivative non-NGB couplings would be

of the form detE (∇aχ†σb←→∇ b∇aχ)|φ|2, with a model dependent coefficient.

Since the model dependent contributions include extra Goldstini beyond those contained

in detE or the metric, the resulting equations of motion and the conserved R-current are

different than those originating from the model independent contributions.

3.3 Embedding Quarks and Leptons

We now take a step towards the SM and identify (some of) the SM fermions as Goldstini

filling R-symmetry multiplets. We still work in the limit where gauge and Yukawa interactions

vanish, but ensure that the strong sector itself respects the symmetries of the SM, i.e. gauge

and flavor symmetries, GSM ≡ GGauge × GFlav, with GGauge = SU(3)C × SU(2)L × U(1)Yand GFlav = SU(3)5Flav × U(1)B × U(1)L.14 Since the only symmetry under which Goldstini

transform is by definition the R-symmetry, GR, the embedding of the SM fermions is possible

only if their gauge and flavor symmetries can be embedded into the R-symmetry, GSM ⊂ GR.

13One could naively think that |φ|2 contributes to the χχ→ χχ elastic Goldstino scattering when φ gets a

VEV, and that there could be a lower bound on how much tachyonic the mass squared can get, from the usual

positivity of the Goldstino amplitude [12]. However, a SUSY-preserving potential for φ must have vanishing

vacuum energy, e.g. V = λ(|φ|2 − f2/2)2, since by construction we include it all in the definition of F in

Eq. (21). This ensures that the χχ→ χχ is actually not affected as long as the potential is SUSY-preserving.14Whenever including the right-handed neutrinos νc we actually consider SU(3)6Flav.

12

3.3.1 Maximal R-symmetry

N free Weyl fermions, which is what the SM reduces to in the limit gSM ≡ {g, g′, gs, Yψ} → 0,

enjoys an U(N) symmetry such that they transform in the fundamental representation. While

gauge interactions in the SM break explicitly this U(45) symmetry down to the smaller GSM ,

the actual representations of quarks and leptons with respect to GSM do not fit a fundamental

representation of the original U(45), simply because the SU(2)L singlet and doublet quarks

come in different color representations, namely a 3∗ and 3 respectively.

These facts may be replicated in the strong sector; the lowest-dimensional interactions (the

Akulov-Volkov Lagrangian Eq.(22)) are accidentally U(N )R symmetric, with Goldstini in the

fundamental representation. Adding higher-dimensional terms, e.g. cklij∇aχi∇aχj∇bχ

†k∇bχ†l ,

generically breaks U(N )R to a subgroup under which the Goldstini transform in various

representations that no longer fit, generically, into the fundamental of U(N )R. However, if

SM matter is to be identified as Goldstini, these representations must fit at the very least the

proper SM representations of GGauge. While it is technically possible that GR = Ggauge (or

GR = GSM) is strictly smaller than U(N ) and that the Goldstini representations are exactly

those of the SM fermions, it would be very surprising, lacking a dynamical reason. With no

better option, we assume instead in the following that the SUSY-breaking sector is maximally

R-symmetric, i.e. GR = U(N )R. The reduction to GSM and then to GGauge is entirely due to

the external SM couplings gSM , rather than the SUSY-preserving parameters of the strong

sector. In other words, we extend the paradigm of MFV from GSM to the largest group that

can be simultaneously preserved by the strong sector and the free theory, i.e. U(N )R.

Interestingly, in most cases the assumption of maximal R-symmetry does not obstruct a

proper embedding of the SM fermions as Goldstini. Maximal R-symmetry is often obtained

when the strong sector has the least number of supercharges compatible with the assigned

Goldstini content, see Table 3. In other words, the representations of the SM fermions often fit

in the fundamental of U(N )R. Assuming maximal R-symmetry is relevant, for what regards

the SM embedding, only when both doublet and singlet quarks are Goldstini, since it requires

a number of supercharges larger than the number of Goldstini-like SM fermions, implying

extra light fermions in the spectrum.

Maximal R-symmetry means, in practice, that we should embed the SM matter represen-

tations that are associated to Goldstini in the fundamental of U(N )R. To do so, the following

decompositions turns out to be useful15

SU(N ×M) ⊃ SU(N)× SU(M) ⇒ = ( , ) , (27)

SU(N +M) ⊃ SU(N)× SU(M)× U(1) ⇒ = ( ,1)a/N ⊕ (1, )−a/M . (28)

15In the first case, an index I in the fundamental that runs from 1 to N ·M can be split into a collective pair of

indexes I = (i, j) where i = 1, . . . N and j = 1, . . .M . In the second case the collective index I = 1, . . . N +M

is split in two separate indexes, i = 1, . . . , N and j = N + 1, . . . , N +M .

13

3.3.2 Embedding Leptons

Let us start with the simplest case: the singlet electron, charged only under hypercharge, ec =

(1,1)1 under SU(3)C × SU(2)L × U(1)Y . In this case we promptly identify the hypercharge

with the U(1)R of N = 1, that is χ = ec.

The case where all three generations of singlet electrons are Goldstini is slightly more

interesting as we insist on a SU(3)ec flavor symmetry. The proper embedding is via a triplet

eeec = (ec, µc, τ c) = 31 of SU(3)R × U(1)R in N = 3, where the U(1)R and SU(3)R factors are

identified with the hypercharge and the flavor group respectively. In this case the R-symmetry

index j is nothing but that the flavor index, χj = ecj.

Embedding only one lepton doublet, ` = (1,2)−1/2 under GGauge, requiresN = 2 and again

maximal R-symmetry U(2)R ∼ SU(2)L × U(1)Y . The R-symmetry index is an electroweak

index in this case, `j = (νL, eL)j = χj. Similarly, including all lepton doublets requires to

considerN = 6 and to decompose the fundamental of U(6)R as 6−1/2 = (2,3)−1/2 with respect

to SU(2)R×SU(3)R×U(1)R. One then identifies SU(2)R×U(1)R with SU(2)L×U(1)Y and

SU(3)R with the flavor group.

With little extra effort we can embed all leptons, including the singlet neutrinos νc, taking

N = 12 and the maximal R-symmetry group U(12)R. Such a large R-symmetry contains

the proper subgroups SU(12)R × U(1)R ⊃ SU(6) × SU(6) × U(1)R × U(1)A that in turn

⊃ (SU(3)× SU(2))×(SU(3)× SU(3)× U(1)C)×U(1)R×U(1)A, where the three SU(3)’s are

identified with the flavor groups acting on the flavor triplets `, eeec and νννc, while the SU(2) factor

is identified with SU(2)L. The last abelian U(1)C allows us to give independent hypercharges

to doublet and singlet leptons. Explicitly, the fundamental of U(12)R decomposes as

12r = (6,1)r,a ⊕ (1,6)r,−a = (3,2,1,1)r,a,0 ⊕ (1,1,3,1)r,−a,c ⊕ (1,1,1,3)r,−a,−c = LLL⊕ eeec ⊕ νννc

under the chain of subgroups we have mentioned above. The hypercharge is identified with

Y = −A/(2a) + C/(2c), the lepton number is L = A/a, while the U(1)R plays no role (we

could have demanded just SU(12)R rather than the maximal U(12)R). We summarize these

and other cases in Table 3.

3.3.3 Embedding Quarks and Extra Exotics

The embedding of either SU(2)L doublet or singlet quarks works like for leptons, as we show

in Table 3. Things become more complicated when we embed the quark doublets together

with the singlets inside the same fundamental representation of SU(N ). This is due to the

difficulty in obtaining both a 3∗ (for dc and uc) and a 3 (for q) of SU(3)C when decomposing

the fundamental of SU(N ).16 The solution to this problem is to add extra states to fill a larger

multiplet that can give rise to 3∗’s, since a 3∗ of SU(3) can be built out of two fundamentals,

16The 36 is the minimal representation that could in principle accommodate the 18 + 9 + 9 quarks qqq, uuuc

and dddc. However, the decomposition of U(36)R into(SU(2)× [SU(3)]2

)×([SU(3)]2 × [SU(3)]2

)× U(1)R ×

U(1)A × U(1)C is 36r = (2,3,3,1,1,1,1)r,a ⊕ (1,1,1,3,3,1,1)r,−a,c ⊕ (1,1,1,1,1,3,3)r,−a,−c which does

not contain a color 3∗ with the other three SU(3)’s identified as the flavor groups.

14

Goldstini GGauge ×GFlav Nminec U(1)Y N = 1

`e SU(2)L × U(1)Y N = 2

`e, ec SU(2)L × U(1)Y × U(1)Le N = 3

`e, ec, νce SU(2)L × U(1)Y × U(1)Le N = 4∗

dc or uc SU(3)C × U(1)Y N = 3

eeec U(1)Y × SU(3)Flave N = 3

` SU(2)L × U(1)Y × SU(3)Flav` N = 6

`, eeec SU(2)L × U(1)Y × U(1)L × SU(3)2Flav N = 9

`, eeec, νννc SU(2)L × U(1)Y × U(1)L × SU(3)3Flav N = 12∗

dddc or uuuc SU(3)C × U(1)Y × SU(3)Flavd(u) N = 9

qqq SU(2)L × SU(3)C × U(1)Y × SU(3)Flavq N = 18

dddc, uuuc SU(2)L × [SU(3)C ]2 × [U(1)Y ]2 × SU(3)2Flav N = 18

qqq,dddc,uuuc,XXX−2/3,1/3 SU(2)L × [SU(3)C ]3 × [U(1)Y ]2 × U(1)B × SU(3)3Flav N = 72 (36)

`, eeec, νννc, qqq,dddc,uuuc,XXX−2/3,1/3 [SU(2)L]2 × [SU(3)C ]3 × [U(1)Y ]4 × U(1)B × U(1)L × SU(3)6Flav N = 84 (48)

Table 3: The minimal supersymmetry Nmin needed to reproduce the correct quantum numbers of SM fields

under gauge and flavor symmetries GGauge × GFlav, including the baryon and lepton numbers U(1)B and

U(1)L. Boldface characters denote flavor triplets. The notation [SU(m)C,L,Y ]n means there are n SU(m)

factors of which only one linear combination corresponds to the gauged SU(m)C,L,Y group. The asterisk ∗

marks whether SU(N )R, as opposed to U(N )R, is required. The number of supercharges inside parenthesis

refers to Nmin for non-maximally R-symmetries (with Goldstini not in the fundamental representation).

3 ⊗ 3 = 3∗ ⊕ 6. Then, the smallest group that can accommodate all quarks (all flavors) is

SU(72)R × U(1)R, with the 72r decomposing as

72r =(2,3,3,1,1,1,1)r,3a,0 ⊕ (1,1,1,3,3∗,1,1)r,−a,c ⊕ (1,1,1,3,6,1,1)r,−a,c (29)

⊕ (1,1,1,1,1,3,3∗)r,−a,−c ⊕ (1,1,1,1,1,3,6)r,−a,−c = qqq ⊕ uuuc ⊕XXX−2/3 ⊕ dddc ⊕XXX1/3

with respect to the subgroup(SU(2)× [SU(3)]2

)×([SU(3)]2 × [SU(3)]2

)× U(1)R × U(1)A × U(1)C .

A diagonal SU(3) out of three SU(3)’s is identified with the color group while three other

SU(3)’s represent the flavor group SU(3)Flavq ×SU(3)Flavd ×SU(3)Flavu (this matter content is

then consistent with two extra global SU(3) factors). The hypercharge reads Y = −R/(12r)+

A/(12a)− C/(2c), whereas the baryon number is B = −R/(6r) + A/(6a).

The prediction of maximal R-symmetry is then that there are extra (pseudo-)Goldstini

X−2/3,1/3 that are colored and charged under Y , transforming as

XXX−2/3,1/3 = (6,1)−2/3,1/3 (30)

under GGauge. From the decomposition in Eq.(29) we see that these exotic states (aka quixes)

are also triplets of the flavor groups, either SU(3)u or SU(3)d, depending on their hypercharge.

15

Similarly, embedding all SM quarks and leptons into Goldstini requires N = 84 super-

charges, where the fundamental splits as 84 = 12⊕ 72, and we can apply the results derived

above. The extra 36 states correspond again to three families of the two exotic color sextets.

There are two interesting points related to this extra matter content. First, they give rise

to [SU(3)C ]2U(1)Y and [U(1)Y ]3 anomalies. In order to cancel them there must exist extra

light colored states, such as complex conjugate color sextets,

YYY 2/3,−1/3 = (6∗,1)2/3,−1/3 , (31)

to form, along with X, real representations of GGauge. The YYY ’s are anti-triplets of flavor. We

will loosely refer to these Dirac fermions Ψ6 = X ⊕ Y c as the sextet 6. The sextet is light,

relative to the strong coupling scale m∗ ∼√g∗F , since a mass term m6XY requires breaking

SUSY explicitly (see section 4). Second, the sextet contributes significantly to the running of

the strong coupling: four extra Weyl fermions (two pseudo-Goldstini X and two extra fields

Y ), in the 6 and 6∗ of SU(3)C , per three generations, imply a contribution δb6 = −20 to the

1-loop β-function β1−loop = −bg3/(16π2). This implies that above m6 the β-function changes

sign and would hit a Landau-pole at roughly Λ ' m6Exp[2π/(bαs(m6))] ≈ 102m6, where we

have taken αs(m6) ≈ 0.09. This sets an upper bound on the scale of strong coupling m∗,

which should enter before Λ. Equivalently, this sets a lower bound on m6/m∗ & 10−2.

4 Explicit SUSY Breaking

So far we have discussed the properties that characterize the strongly interacting sector, in

different cases where one or more SM fermions are Goldstini, but in the limit of vanishing

SM interactions gSM → 0. The leading interactions genuine of goldstino-compositeness are

controlled by dimension-8 operators, which are large at high energy, but vanishingly small at

small energy. These interactions preserve a maximal R-symmetry, which contains the relevant

global groups to be gauged and flavor groups.

This picture is necessarily distorted at least by the marginal SM interactions, which are

small but become leading at sufficiently small energy. These interactions break SUSY explic-

itly, the Goldstino-like SM fermions become pseudo-Goldstini, and new effects are generated,

which we estimate in this section.

The SM interactions should be thought as spurions of SUSY breaking,

∆Lbreak = −Yχ ij[χiχjH + . . .

]+ h.c.+ gV

[V Aµ R

Aµ + . . .]. (32)

The Yukawas Y also break the flavor symmetries and we assume these are the only sources

of flavor breaking, thus realizing the MFV paradigm [36]. Gauge interactions arise through

weakly gauging some of the R-currents RAµ, Eq. (26). The dots in Eq. (32) represent the

generalization of the minimal symmetry-breaking interactions (the first terms in brackets) by

operators with the same field, symmetry, and spurionic content but with extra (covariant)

derivatives, suppressed by m∗.

16

Fermion-Gauge Fermion-Higgs

OψB = DνBµν(ψL,RγµψL,R) OψL,R = (iH†

↔DµH)(ψL,Rγ

µψL,R)

OψW = DνW aµν(ψLσ

aγµψL) O(3)ψL = (iH†σa

↔DµH)(ψLσ

aγµψL)

Dipoles Oyψ = |H|2ψLHψRODB = ψLσ

µνHψRBµν Four-Fermions

ODW = ψLσµνσaHψRW

aµν O4ψ = ψγµψψγ

µψ

ODG = ψLσµνHTAψRG

Aµν

Table 4: Dimension-6 operators involving fermions relevant for our analysis.

We focus first on the scenario where the only light composite d.o.f.’s are the pseudo-

Goldstini, while non-Goldstini SM fermions, the Higgs and the gauge bosons are elementary;

we comment below on extensions of this picture. In such minimal scenario no dimension-

6 operators are generated directly by the strong dynamics, but some might be generated

via loops involving both the strong dynamics and the spurions in Eq. (32) – a list of the

interesting dimension-6 operators is reported in Table 4. We estimate these effects by simple

power counting, derived from the leading loop diagrams, some of which are shown in Fig. 2.

Z couplings

The largest effects involve operators with external fermions and the least number of SUSY-

breaking spurions. In particular, loops of pure strong sector with a gauge boson insertion

generate OψW,B (see e.g. Fig. 1(a)) with coefficients

cψW,B ∼g

m2∗. (33)

Through a field redefinition (which corresponds to the equations of motion) these operators

are equivalent to combinations of OψL,R, O(3)ψL and O4ψ, with coefficients cψL,R ∼ c

(3)ψL ∼ g2/m2

∗and c4ψ ∼ g2/m2

∗. The former modify the couplings of the Z boson to fermions, which

are constrained at the permille level from measurements at LEP-1. Consequently, goldstino-

compositeness of doublet leptons (or L-handed, `L, in Dirac notation), is constrained as [45,46]

` Goldstini: m∗ & 2.5 TeV , (34)

while goldstino-compositeness of singlet leptons (or R-handed, eR), even if probed by the same

data, gets a milder constraint due to the (g′/g)2 suppression of the corresponding operators

eeec Goldstini: m∗ & 2 TeV . (35)

We will see in section 5.2 that, because of the low energies accessible at LEP, the constraints

on pseudo-Goldstini leptons from their defining dimension-8 operators are weaker than the

indirect ones derived here.

17

ψ

ψ

g V

(a)

ψ

ψ

Y

Y

H†

H

(b)

ψ

ψ

ψ

ψ

g

g

(c)

Figure 1: Some leading diagrams contributing to dimension-6 operators with fermions. Black blobs denote

strong interactions, while little red (blue) ones denote SM gauge (Yukawa) couplings that break SUSY explicitly.

Given the constraints on modifications of the L-handed quark couplings to gauge bosons,

for instance from LEP measurements of the Z decay to hadrons, their goldstino-compositeness

is indirectly constrained at the same level as that of L-handed leptons

qqq Goldstini: m∗ & 2.5 TeV . (36)

On the contrary, goldstino-compositeness of R-handed quarks is not very constrained via the

effects of Ou, dR , given that the sensitivity of LEP to non-standard R-handed quark couplings

is significantly reduced by their small SM coupling to the Z.

The only relevant Yukawa-induced operators are those associated to up quarks as pseudo-

Goldstini, because of the large top Yukawa. In particular, if both L- and R-handed tops are

pseudo-Goldstini, we find (from e.g. Fig. 1(b)) an additional contribution to

c(3)ψL ∼ cψL,R ∼

Y 2t

m2∗. (37)

This could be the dominant effect, since it modifies the Zbb coupling, implying

q3 and tc Goldstini: m∗ & 5 TeV . (38)

However, if q3 is elementary this bound disappears, while if tc is elementary, loop diagrams like

Fig. 1(b) become weak and thus suppressed by g2∗/16π2, implying c(3)ψL ∼ cψL ∼ Y 2

t g2∗/16π2m2

∗,

which might alleviate the bound if the strong coupling is not maximal, g∗ < 4π. Other

constraints associated with Eq. (37) come from top physics only and therefore are mild.

Higgs couplings

The operators Oyψ are generated with coefficients cyψ ∼ Y 3ψ /m

2∗ (or Yψg

2g2∗/16π2m2∗) given

that the discrete symmetry H → −H is broken only by the Yukawas. The experimental

constraints from measurements of the Higgs couplings to fermions are not competitive enough

to make these effects relevant. Furthermore, this estimate is reduced by g2∗/16π2 if only the

L- or R-handed fermions are pseudo-Goldstini.

18

ψR

ψL

Yg

H

V

(a)

ψ

ψ

ψ

ψ

Y

Y

(b)

Figure 2: Some leading diagrams contributing to (magnetic) dipole moments and flavor transitions.

Four-fermion contact interactions

The operators O4ψ can be generated with coefficients c4ψ ∼ g2/m2∗ from the field redefini-

tion described below Eq. (33), or directly from diagrams like Fig. 1(c), carrying an additional

g2∗/16π2 suppression due to the elementary vector boson in the loop. These operators con-

tribute to the same observables as the bona fide Goldstini operators of dimension-8, Eq. (24).

We leave the analysis of these for section 5.

Dipole Moments

When both L- and R-handed SM fermions are pseudo-Goldstini, a Higgs and a gauge inser-

tions are enough to saturate the necessary selection rules to generate dipole operators with

V = B,W,G (see e.g. Fig. 2(a)),

ψψψL and ψψψR Goldstini: cDV ∼YχgVm2∗. (39)

Instead, if only ψL or ψR are composite, dipole operators can only be generated upon insertion

of two extra Yukawa couplings and an elementary Higgs loop similar to the Barr-Zee one [47],

ψψψL or ψψψR Goldstini: cDV ∼Y 3χ gV

(4π)2m2∗. (40)

The precise measurement of the anomalous magnetic moment of the muon [48] sets a con-

straint on goldstino-compositeness of leptons

` and eeec Goldstini: m∗ & 3.2 TeV (41)

while if only ` or eeec are pseudo-Goldstini the bound is negligible.

We note that our assumption on R-symmetry, which provides a concrete realization of the

MFV paradigm, implies no new sources of CP violation beyond the SM. Therefore we obtain

no relevant constraints from electric dipole moments.

Flavor transitions

In MFV any flavor-violating operator is proportional to the SM Yukawa couplings. This means

that in the lepton sector any flavor transition is negligible due to the smallness of neutrino

19

masses. In the quark sector instead, MFV implies that any flavor-violating process proceeds

via the top, and the scale associated to certain operators needs to be above 1 TeV [49]. Some

of the potentially problematic operators, e.g.

e

(4π)2m2∗dRY

†d YuY

†uσ

µνH†qLFµν ,g2∗

(4π)4m2∗(qLYuY

†u γµqL)2 , (42)

are generated at one or two weak (Higgs) loops respectively, so that the sensitivity of observ-

ables such as B → Xsγ or εK and ∆mBd is suppressed. The strongest flavor constraints on

goldstino-compositeness of quarks arise instead from the operator (see e.g. Fig. 1(b))

1

m2∗(iH†

↔DµH)(qLYuY

†u γ

µqL) (43)

that contributes to B → Xs`+`−, Bs → µ+µ−, giving rise to the bound

qqq and uuuc Goldstini: m∗ & 2.3 TeV , (44)

similar in size to the one from electroweak precision tests Eq. (36).

Comparable constraints arise if both L-handed quarks and leptons are pseudo-Goldstini.

In such a case diagrams like Fig. 2(b) generate the operators

g2∗(4π)2m2

∗(qLYuY

†u γµqL)(¯

Lγµ`L) ,

g2∗(4π)2m2

∗(qLYuY

†u γµqL)(eRγ

µeR) , (45)

which also contribute to B-decays, implying

qqq and (` or eeec) Goldstini: m∗ & (1.7 or 2.7 TeV)×( g∗

4π

). (46)

Other couplings

Operators without external fermions can also be generated, simply because of the existence

of a sector with a mass scale m∗ to which all SM fields couple. Yet, such operators do not

benefit from any strong coupling enhancement and provide, therefore, less relevant sensitivity

to scenarios where (some) fermions are pseudo-Goldstini but the other species are elementary.

Other composites

It is certainly plausible that, if SM fermions are indeed pseudo-Goldstini, the strong dynamics

also involves in some way the Higgs boson (perhaps in a solution to the hierarchy problem),

the transverse polarizations of vectors, or perhaps it includes some of the fermions as non-

Goldstini composites. These scenarios have been studied extensively already, in e.g. Refs. [1,3,

7], and their predictions in terms of EFTs are well known: composite fermions imply largeO4ψ,

as already discussed in this work, while composite Higgs models imply large OH = (∂µ|H|2)2and models of composite gauge vectors are characterized by large O2W = (DρW

aµν)

2 and

O3W = εabcWa νµ W b

νρWc ρµ.

20

In this context, if the Higgs is composite the size of some of the Yukawa-induced operators

discussed above is enhanced. In particular, operators that were generated via an elementary,

thus weak, Higgs loop, see e.g. Fig. 2(b), are no longer suppressed by g2∗/16π2, since for a

composite Higgs the loop becomes strong.17 Such an enhancement applies to Eqs. (40) and

(42), (45). Similarly, operators that were suppressed by g2∗/16π2 because either ψL or ψR were

not pseudo-Goldstini, are enhanced as well when the Higgs is composite.

An interesting case is that in which the low-energy EFT includes non-SM light composite

states, for which the selection rules differ. We have discussed in section 3.3 that embedding

all of the quarks as Goldstini requires the existence of new color sextets X and Y . Masses for

these states require an extra source of explicit SUSY breaking, which we simply write as

m62/3X−2/3Y2/3 +m61/3

X1/3Y−1/3 . (47)

Being this a small deformation of the SUSY-preserving dynamics, we expect the sextets to be

naturally light, thus present in the low-energy spectrum. Furthermore, the pseudo-Goldstini

X have SUSY-preserving interactions which, because of maximal R-symmetry, are invariant

under X → −X. Therefore such interactions do not contribute to the sextet’s decay to the

SM: symmetry breaking effects can play a major role here. Specifically, a new SUSY-breaking

spurion (with different quantum numbers than gSM or m6) can be introduced in association

with a linear coupling of the sextet, the lowest-dimensional one being, schematically

εgsm∗

ΨσµνχGµν , (48)

where Ψ = Y2/3,−1/3 and χ = uc, dc, where recall Y is a non-Goldstini state. In fact, Y could

well be external to the strong dynamics, in which case its degree of compositeness should be

factored in. Besides, ε � 1 is expected since (48) is an irrelevant operator. This operator is

not only important for the sextet decay but also for its single production at the LHC, as we

discuss in section 6.

5 Collider Phenomenology

In section 3 we have built the Goldstini EFT in the symmetric limit: Goldstini are char-

acterized by interactions of dimension-8 that grow maximally fast with energy. We have

further established in section 4 that the less irrelevant dimension-6 operators can only arise

from (suppressed) SUSY-breaking effects. Constrains on the latter imply m∗ & 2 TeV for the

compositeness scale in most of the interesting cases (with the exception of R-handed quarks,

where the bounds are weaker).

In this section we attempt instead direct access to the strongly coupled dynamics by con-

sidering 2→ 2 pseudo-Goldstini scattering at the highest possible energies. For comparison,

17This can also be understood by identifying, when the Higgs is composite, the Yukawa coupling in Eq. (32)

with a linear mixing a la partial compositeness, see e.g. Ref. [7].

21

we study both our scenario based on non-linearly realized SUSY and dimension-8 operators,

as well as the standard one based on dimension-6 operators, which updates the analysis of

Ref. [3]. Our main focus is LHC phenomenology, which gives us access to scattering of quarks

in the multi-TeV region, but we also brush upon LEP-2 to investigate the extend to which

leptons could arise as pseudo-Goldstini.

The results on composite quarks are summarized in Table 5: we show the bounds on

the SUSY-breaking scale√F for goldstino-compositeness, as well as those on the scale f for

chiral-compositeness, in different composite quark scenarios. A direct comparison between

the two types of compositeness is also presented in terms of bounds on the strong coupling

scale, identified as m∗ ≡√g∗F for Goldstini and m∗ ≡ g∗f for chiral composites. The results

on composite leptons, specifically eR and µR, are given in Eqs. (58) and (59).

5.1 LHC

We focus on the process that gives, at present, access to the highest energies: dijet events

pp → jj initiated by valence quarks at 13 TeV. The experimental analysis can be found in

Ref. [50], corresponding to an integrated luminosity of 15.7 fb−1 collected with the ATLAS

detector. The relevant pseudo-Goldstini interactions are parametrized by the operators

Ouu = (∂ν uRγµuR)(uRγµ∂

νuR) Oud = (∂ν uRγµuR)(dRγµ∂

νdR) + h.c.

Odd = (∂ν dRγµdR)(dRγµ∂

νdR) Oqu = −(∂ν qLγµqL)(∂ν uRγµuR) + h.c.

Oqq = (∂ν qLγµqL)(qLγµ∂

νqL) Oqd = −(∂ν qLγµqL)(∂ν dRγµdR) + h.c. , (49)

where in each particular composite quark scenario the Wilson coefficients are cij = 1/(2F 2).

These operators contribute to the differential dijet cross section at the partonic level,18

dσ

dt(qiqi → qiqi)BSM =

4αs9Aqi1 +

1

48π

[Bqi

1 (2u2 + 2t2 + s2) + 6Bqi2 (u2 + t2)

],

dσ

dt(ud→ ud)BSM =

1

16πu2B3 , (50)

where s, t, u are the Mandelstam variables (s+ t+ u = 0 in the massless approximation) and

Au,d1 = cqq + cuu,dd , Bu,d1 = c2qq + c2uu,dd , Bu,d

2 = c2qu,qd , B3 = c2qq + c2ud + c2qu + c2qd .

The BSM contribution is then characterized by a strong energy growth and by being more

central (−t ≈ s) than the SM one, which instead peaks in the forward region because of t-

channel gluon exchange. The analog of the CM scattering angle θ is conveniently represented

by the boost-invariant difference between the two jet rapidities yj = log√

(Ej + pzj)/(Ej − pzj),

χ ≡ e|yj1−yj2 | = (1 + cos θ)/(1− cos θ) . (51)

18The leading contributions at high energies come from two initial first-family quarks (as opposed to quark-

antiquark, two antiquarks or second- and third-family quarks), due to the enhanced PDFs in pp collisions.

22

1 2 5 10 20

0.02

0.04

0.06

0.08

0.10

● ●● ● ● ● ● ● ● ●

●

■ ■■ ■ ■ ■ ■ ■ ■ ■

■● ●

● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

●●

●●

●● ● ●

● ●●

■

■

■

■

■

■ ■ ■■ ■

■

1 2 5 10 200

10

20

30

40

50

Figure 3: Left: Dijet angular distributions in the highest-energy bin Mjj > 5.4 TeV, for the experimental

data with its systematic plus statistical uncertainties (black points), the SM prediction with its theoretical error

(blue band), and the BSM predictions for the dimension-8 operator Odd (solid orange) or the dimension-6

operator O(1)dd [3] (dashed red). The corresponding (positive) coefficients have been chosen to saturate the

bounds Eqs. (52) and (53). Right: Bounds on the scale m∗ for a composite dR with a dimension-8 operator

|cdd| = (4π)2/2m4∗ (orange) or a dimension-6 operator |c(1)dd | = (4π/m∗)

2 (red) and for all quarks composite

with dimension-8 operators |cij | = (4π)2/2m4∗ (black), for different χ bins (dots) and for all bins combined

(lines). The round dots and solid lines correspond to c > 0 while square dots and dashed lines to c < 0.

The central region corresponds to small χ, where the BSM differential cross section peaks,

while the SM is approximately flat. Such a behavior can be recognized in Fig. 3 left panel,

where we show the SM distribution and two different BSM contributions both corresponding

to a composite dR, one from goldstino-compositeness (cdd 6= 0) and the other from chiral-

compositeness (c(1)dd 6= 0, operators and cross section formulae can be found in Ref. [3]).

We compute the BSM particle-level cross section in two different ways to double-check:

via the analytic differential cross section Eq. (50), integrated over the PDFs (CT10 [51]) and

binned according to the experimental analysis, and via a MadGraph [52] simulation of our

model implemented with FeynRules [53]. This is compared with data from Ref. [50], limited

to highest-energy bin, with an invariant mass Mjj > 5.4 TeV; we check, a posteriori, that

the bounds obtained on the compositeness scale m∗ are compatible with the EFT hypothesis:

m∗ �Mjj. The analysis includes additional cuts on the transverse momentum of the leading

jet pT (j1) > 450 GeV, on the average rapidity of the two jets (yj1 + yj2)/2 < 1.1, and on the

rapidity difference |yj1 − yj2|/2 < 1.7. For the SM prediction we use the NLO differential

distribution reported in Ref. [50], but we normalize it to the total number of SM events,

compatible with the above cuts, that we compute using POWHEG [54] and PYTHIA8 [55]

with PDF4LHC15 [56], obtaining σcutsSM = 50.8± 9.1 fb (corresponding to ∼ 800 events for the

integrated luminosity of Ref. [50]). NLO effects on the new physics distribution are instead

neglected, although these could be relatively important compared to SM NLO effects, in the

region of parameters that saturates the bounds (in this region, the SM and BSM contributions

are similar in size, see Fig. 3).

Concerning the errors in our analysis, in the small χ region the dominant uncertainty is

23

goldstino chiral

composites√F [m∗] (TeV) f [m∗] (TeV)

dR 2.6 [9.4] 2.9 [36]

uR 3.8 [13.5] 4.7 [59]

uR, dR 3.9 [13.7] 4.9 [62]

qL 3.9 [13.7] 4.9 [62]

qL, dR 4.0 [14.2] 5.0 [63]

qL, uR 4.5 [16.1] 5.7 [72]

qL, uR, dR 4.6 [16.2] 5.8 [73]

Table 5: 95% CL bounds on the SUSY-breaking scale√F for different quark as pseudo-Goldstini scenarios

(second column) and on the scale f for different cases of chiral-compositeness of quarks (third column). The

respective Wilson coefficients are given by cij = 1/(2F 2) for the operators in Eq. (49) and by c(1)ij = 1/f2 for

the operators O(1)ij = (ψiγ

µψi)(ψjγµψj) given in Ref. [3]. The compositeness scale in each case is given by

m∗ ≡√g∗F or m∗ ≡ g∗f with g∗ = 4π, their bounds reported in brackets.

statistical, since the SM cross section is dominated by the forward region and therefore in

the central region the statistics is small. Systematic uncertainties (dominantly from the jet

energy scale) are relatively small. Theoretical uncertainties are sizable, yet smaller than the

statistical at small χ, becoming the dominant uncertainty at large χ. Besides, even though

the errors of the SM cross section are not gaussian, we treat them as such to simplify the

analysis, symmetrizing the distributions such that the 1σ band remains unaltered.

We extract bounds on the Wilson coefficients in each χ bin and combine them to obtain

the final constraint. In Fig. 3 we report these bounds for the case of a composite dR as a

function of m∗, defined as cdd ≡ ±g2∗/2m4∗ for goldstino-compositeness and c

(1)dd = ±(g∗/m∗)

2

for chiral-compositeness, with strong coupling g∗ = 4π. The final bounds for positive (m+∗ )

or negative coefficients (m−∗ ) at the 95% CL are19

dR chiral composite: m+∗ & 36 (g∗/4π) TeV , m−∗ & 40 (g∗/4π) TeV . (52)

dR Goldstini: m+∗ & 9.4

√g∗/4πTeV , m−∗ & 9.0

√g∗/4πTeV . (53)

Recall these bounds are derived from the bin with Mjj > 5.4 TeV. This means that for our

analysis to be safely consistent with the EFT assumption Mjj � m∗, we must require g∗ & 2

in the case of chiral-compositeness and g∗ & 4.5 for goldstino-compositeness, i.e. the new

physics must be strongly coupled.

Besides, we discussed in section 4 the existence of dimension-6 four-fermion operators

O4ψ generated from explicit SUSY-breaking. The question arises then whether experimental

searches, in the case at hand 2 → 2 quark scattering at the LHC, are more sensitive to

these more relevant (but gSM -suppressed) effects or to the dimension-8 (but g∗-enhanced)

19Goldstino operators are in fact subject to positivity constraints that require strictly positive coefficients

(see section 5.3). Our bounds for c < 0 simply illustrate the relevance of the interference with the SM.

24

effects. Eqs. (52) and (53) show that, for g∗ within the validity of the EFT, the constraints on

pseudo-Goldstini from the latter are clearly superior. As a matter of fact, bounds on goldstino-

compositeness from the LHC are stronger than any indirect ones from SUSY-breaking effects.

This is clearly due to the large energies accesible at this collider (roughly E/m∗ � gSM/g∗).

Finally, the scenario of a composite dR is constrained the weakest of all the cases, while

when all quarks are composite, the LHC reach is maximal. In Fig. 3 we show the constraints

in this scenario, the final result being

qL, uR, dR Goldstini: m+∗ & 16.2

√g∗/4πTeV , m−∗ & 14.2

√g∗/4πTeV . (54)

The bounds for the rest of composite scenarios are given in Table 5.

5.2 LEP

During its second phase, LEP collided electrons and positrons at energies significantly higher

than the Z-pole, measuring angular distributions with percent precision. Here we focus on

how the differential cross sections for e+e− → e+e−, µ+µ− are affected by the goldstino-

compositeness of the R-handed electron eR and muon µR, and compare them with LEP-2

data [57] to extract bounds on their SUSY-breaking scale F or compositeness scale m∗.

Let us first recall that limits on standard chiral-compositeness of eR, parametrized by the

dimension-6 operator c(6)(eRγµeR)(eRγµeR), were extracted by the LEP collaborations,

eR chiral composite: m+∗ & 43 (g∗/4π) TeV , m−∗ & 40 (g∗/4π) TeV , (55)

where we normalized the corresponding Wilson coefficient as |c(6)| = (g∗/m∗)2. Bounds were

obtained as well on (eRγµeR)(µRγµµR): m+

∗ & 41 (g∗/4π) TeV and m−∗ & 33 (g∗/4π) TeV.

Our interest here is in deriving first-time bounds on the goldstino-compositeness of eR,

which is parametrized by the dimension-8 operator

Oee = (∂ν eRγµeR)(eRγµ∂

νeR) , (56)

and on the scenario where both eR and µR are pseudo-Goldstini, in which besides Oee also

Oeµ = (∂ν eRγµeR)(µRγµ∂

νµR) + h.c. (57)

is generated, with the same coefficient.20 These operators induce non-standard contributions

to the differential cross sections for Bhabha scattering and dimuon production – their analytic

expressions are reported in the Appendix. They share some similarities with the dijet case,

in particular the t-channel photon exchange also gives rise to a forward singularity, such that

the SM contribution and the interference with the BSM are enhanced for small angles.

The experimental sensitivity in e+e− production at small angles is approximately 4%

(95% CL) of the SM contribution. This means that, contrary to the LHC, LEP is really

20We do not study τR compositeness since the sensitivity of LEP in e+e− → τ+τ− is weaker.

25

testing small departures from the SM, and it is sensitive then to the SM-BSM interference

term. We take the SM prediction provided in Ref. [57] and compute the new physics effects

analytically, see the expressions in the Appendix. The theoretical uncertainties on the total

SM cross section amount to 2% for σ(e+e−) and 0.4% for σ(µ+µ−), resulting in an uncertainty

for dσ/d cos θ|SM of 4% and 1% respectively (assuming the error to be uniformly distributed

with the scattering angle θ). For the purpose of this analysis, we use samples of events with

integrated luminosity of 3 fb−1 and increasing effective CM energy from 189 to 207 GeV.21

We combine the limits from different energy and angular bins and obtain

eR Goldstini: m+∗ & 1.8

√g∗/4πTeV , m−∗ & 1.4

√g∗/4πTeV . (58)

eR, µR Goldstini: m+∗ & 1.9

√g∗/4πTeV , m−∗ & 1.5

√g∗/4πTeV . (59)

The bounds on the scenario where both electron and muon are pseudo-Goldstini is driven by

Bhabha scattering, with a SM cross section significantly larger than dimuons – the constrains

on only Oeµ are milder m+∗ & 1.6

√g∗/4πTeV and m−∗ & 1.5

√g∗/4πTeV. It is amusing to

find that goldstino-compositeness of leptons is allowed at incredibly small scales from direct

searches, even for light leptons and maximally strong coupling g∗ = 4π.

In fact, precision is relatively less important, compared to the collider energy, when search-

ing for this type of compositeness, as it becomes clear when comparing the reach of LHC vs

LEP. In contrast, dimension-6 operators are typically better constrained by LEP (or barely

so by the LHC [10]). Indeed, the relative size of our effects scales as δ ∼ (E/m∗)4 in the linear

regime, and as (E/m∗)8 for large deviations from the SM. In order to increase the bound on

the scale m∗ by a factor of 2 we would need to increase the precision at a given energy by

at least a factor of 16; the same goal can be achieved by a factor of 2 increase in energy.

The LHC high-energy reach makes it the best machine to test for fermions with enhanced

soft behavior. Another consequences of this same fact is that the limits on the scale m∗ for

chiral-compositeness are much higher, their effects scaling as (E/m∗)2. This also means that,

for a proper extraction of the bounds, SUSY-breaking effects which generate four-fermion

operators should be included in the analysis [58], even if suppressed by (gSM/g∗)2.

5.3 Positivity Constraints

An interesting aspect of the dimension-8 interactions studied in this work is that they are

subject to positivity constraints. Indeed, the basic requirement of unitarity in the underlying

theory, together with analyticity of the 2→ 2 scattering amplitudes, implies that the Wilson

coefficients of the operators in Eqs. (49) and (56), (57), be strictly positive [12,14].

From a phenomenological perspective, this represents an important prior, from first princi-

ples, to our statistical analysis, that reduces the parameter space by half. Without any prior,

our analysis above leads to 95% CL intervals of the form [−c−(g∗,m−∗ ), c+(g∗,m

+∗ )], while our

21Initial-state photon radiation may reduce the CM energy of the dilepton production. In the LEP analyses

only events with soft initial-state radiation are retained [57].

26

theory prior implies ]0, c(g∗,m∗)]. Taking it into account in our statistical analysis, we find,

dR Goldstini: m∗ & 9.3√g∗/4πTeV , (60)

eR Goldstini: m∗ & 1.7√g∗/4πTeV . (61)

Note that these limits do not improve our knowledge on goldstino-compositeness (compared to

Eqs.(53) and (58)). On the contrary, these more conservative bounds highlight the importance

of keeping the prior, not to overexclude the physically consistent region of parameter space.

Besides, whether c(g∗,m∗) < c+(g∗,m+∗ ) depends on the likelyhood L being symmetric or

not under reflection of the Wilson coefficients c→ −c. Indeed, if it is symmetric then

0.95 =

∫ c+c−dcL∫∞

−∞ dcL=

∫ c+0dcL∫∞

0dcL

, (62)

where the last expression defines the bound with our prior, thus c = c+. Under our assumption

of symmetric errors, a significant asymmetric likelyhood can arise if two conditions are met:

an under or upper fluctuation in the data is present and the size of the constraints is such

that the cross section has a sizable interference term (linear in the Wilson coefficient). In our

LHC analysis, departures from the SM are not large (apart from the third χ bin in Fig. 3).

Most importantly, the experimental resolution, limited by statistical errors at present, is

not enough to resolve the SM-BSM interference term in the cross section, which is instead

dominated by the quadratic new physics contribution.22 For this reason, positivity constraints

do not improve sizably our constraints on m∗ and g∗.

5.4 Outlook - Dibosons

In the previous sections we have shown that goldstino-compositeness is described by dimension-

8 operators, and that currently the LHC is more sensitive to such strongly coupled effects than

to SUSY-breaking effects, even though the latter give rise to lowest-dimensional interactions.

If other species in the SM are involved in the strong dynamics, there can also be large

effects in other LHC processes, beyond 2 → 2 fermion scattering, that are however unique

to pseudo-Goldstini. Interestingly, these effect are characterized by dimension-8 operators as

well, as shown in Tables 1 and 2. Indeed, if the Higgs is composite in addition to the light

quarks, the operator1

2F 2

(iψγ{µ∂ν}ψ + h.c.

)∂µH

†∂νH (63)

modifies the amplitudes for h pair production, as well asW+LW

−L and ZLZL (L = longitudinal).

Similarly, if the transverse (T) polarizations of vectors are strongly interacting, along the lines

of Ref. [7], the amplitudes for W+T W

−T , ZTZT , ZTγ and γγ are modified by

− 1

4F 2FAµνF

Aµρ

(iψγ{ρ∂ν}ψ + h.c.

). (64)

22Notice that this fact is safely compatible with the validity of our EFT expansion, due to the underlying

strong coupling, see the discussion below Eq. (53) and Ref. [8].

27

This is very interesting because, even in a completely model-independent approach, the am-

plitudes for processes with neutral gauge boson final states are not modified at the level of

dimension-6 operators; experimental constraints from ZZ final states are at present already

derived in terms of dimension-8 operators of the form iH†↔DµHD

νBνρBµρ. This kind of op-

erators are typically subleading in theories without symmetries and therefore these type of

searches have received so far little attention. Pseudo-Goldstini offer instead a context where

all dimension-6 operators are naturally suppressed by symmetries and dimension-8 effects are

naturally leading, so that these searches could play the most important role.

Another reason why the operators we propose in this article are interesting is the following.

Currently the entire new physics parametrization of neutral diboson final states, see Ref. [59],

only induces final states with one longitudinal and one transverse vector.23 The corresponding

amplitudes decrease at high energy as 1/E compared to the amplitudes for TT or LL final

states, and is therefore typically subdominant. Instead, the effects we advocate here modify

all TT and LL amplitudes (including the one with (+,−) helicity that dominates in the SM)

which, beside being more generic, will also be easier to find.

In summary, SM fermions as pseudo-Goldstini provide the first structurally motivated

scenario where processes with neutral gauge boson pair production can be used as valuable

BSM search tools. Importantly their parametrization departs from that traditionally adopted

in these searches, and involves a richer variety of phenomena with enhanced high-energy

behavior and larger SM-BSM interference. We leave this for future work.

6 Phenomenology of the new colored states

In section 3, we have found that models with all quarks as pseudo-Goldstini and maximal

R-symmetry require the existence of new exotic colored particles and their extra (non-SM

fermions and not pseudo-Goldstini) partners. Their quantum numbers under the gauge and

flavor symmetries U(1)Y × SU(2)L × SU(3)C × SU(3)Flavu × SU(3)Flavd are

XXX− 23

= (1,6,3,1)− 23, XXX 1

3= (1,6,1,3) 1

3, YYY 2

3= (1,6∗,3∗,1) 2

3, YYY − 1

3= (1,6∗,1,3∗)− 1

3

(65)

They have the same quantum numbers as uc and dc except they transform as a 6 of SU(3)C .

Their Dirac masses m62/3,1/3are naturally small, as they break SUSY explicitly, see Eq. (47).

In the following we will always omit the hypercharge of X and Y , being always understood

its value from the coupling to the SM quarks.

23This is due to Ref. [59] providing a parametrization for anomalous neutral triple gauge couplings that

limit the process to take place through the s-wave, which forbids in this case identical bosons in the final

state; in our case higher waves are allowed and different final states open.

28

Decays

These sextets can couple to gluons and the SM u or d quarks through the model-dependent

SUSY-breaking couplings in Eq. (48) (recalling that 6⊗3 = 8⊕10 and 3⊗8 = 3⊕6∗⊕15)(εgsm∗

)K

bA

i Y iσµνqbGAµν + h.c. , q = uc, dc (66)

where KbA

i are the Clebsch-Gordan coefficients linking the anti-sextet to the anti-triplet and

the adjoint, normalized such that

KbA

i KjA b = δ ji , K

bA

i KiA b = 6 . (67)

ε is a SUSY-breaking parameter that depends on assumptions like the degree of compositeness

of Y . Despite the interactions (66) being suppressed, they represent the main decay mode for

the sextet. In particular, they open the following decay channels:

X, Y → jj, bj, tj (68)

with decays widths approximately given by,

Γ(Y → qg) ≈ αsε2 m

36

4m2∗. (69)

We assume, consistently with ε being a small symmetry-breaking parameter, that Γ � m6,

such that the narrow-width approximation applies to the searches described below.

Production and direct searches

The sextet can either be singly produced via gq → Y (see e.g. Ref. [60]) or pair produced

gg, qq → XX, Y Y . We assume m62/3= m61/3

but focus on direct searches of a single sextet

flavor, coupled to first generation quarks (for a fully degenerate family spectrum, our analysis

can be appropriately rescaled). The relative sensitivity of single production is larger for heavy

sextets, although it requires large couplings ε/m∗, while double production presents a poorer

mass reach, but it is more model-independent given the cross section is fixed by QCD.

The LO partonic cross section for single production and decay is given by

σgq→Y→jj = σgq→Y · BR(Y → jj) , σgq→Y =π2αs

2

(ε

m∗

)2

m26 δ(s−m2

6). (70)

where q = u, d. This cross section suffers from significant NLO corrections (e.g. from gg → Y j)

that we neglect given the exploratory nature of our analysis. We compare our LO prediction,

appropriately convoluted with the PDFs, with experimental searches of singly produced exotic

quark-like resonances decaying into dijets [50,61].

Pair production gives rise to a four-jet signal. We compute the associated cross section at

LO with MadGraph [52], requiring four leading jets with pT > 80 GeV and pseudo-rapidity

|η| < 1.4. We compare our results with the cross section bounds provided by Ref. [62].

29

15.7 fb

95%CL upper limitsingle production

-1

37 fb-1

95%CL upper limitpair production

36.7 fb-1

-

-

-

0 10 20 30 40 50 600

1

2

3

4

5

6

7

Figure 4: Left: 95% CL upper limits on single- and pair-production cross sections, compared with the the-

oretical cross sections for single-production (blue) with m∗/ε = 20 (solid) or m∗/ε = 10 (dashed) and pair-

production (dot-dashed red). Right: Exclusion regions at 95% CL from single-production (dark grey; [50] solid

line and [61] dashed line) and double-production (light grey; dot-dashed line). Also shown are lower bounds

(red) on the (inverse) coupling of the sextet, m∗/ε ≈ 19 TeV (dashed; corresponding to a scale m∗ = 5.4 TeV,

the highest energy of our dijet analysis, and a small parameter ε = 0.3), m∗/ε ≈ 27 TeV (dot-dashed; same

m∗, ε = 0.2) and m∗/ε ≈ 33 TeV (solid; coupling below which pair production dominates the constraints).

We present in Fig. 4 left panel single- and pair-production cross sections along with their

corresponding experimental limits. For single production we take two different values of the

(inverse) coupling m∗/ε = 10, 20 TeV in order to illustrate the variation of the cross section.

In the right panel we show the final bounds in the (m∗/ε,m6) plane. As anticipated, single

production dominates the constraints at high masses and large couplings (i.e. small m∗/ε).

Note that given the sextet is predicted upon all quarks being pseudo-Goldstini (and maximal

R-symmetry), for which LHC searches in section 5.1 set a bound on the compositeness scale

m∗ & 16.2√g∗/4πTeV (see Table 5), the coupling ε/m∗ cannot be too large while keeping

ε � 1. This is exemplified by the vertical (red) lines in Fig. 4, which should be understood

as upper bounds on the sextet linear coupling, restricting the regime where limits from single

production apply. In contrast, pair production sets the robust lower bound m6 & 1.2 TeV.

7 Conclusions and Outlook

The quest for an answer to the question “what is matter made of?” has always been at the

heart of particle physics research. To keep pursuing this endeavor, the priority of the future

high-energy experimental program, both at the High-Luminosity LHC and future colliders, is

to understand to what extent those that we call SM particles are indeed elementary entities,

or to find signs of their substructure.

30

Unitarity of the underlying theory implies there are only two relativistic effective field

theories for fermion compositeness, which can be broadly differentiated from the point of view

of a long-distance observer. The first such scenario is characterized by a series of effective four-

fermion operators of dimension-6, whose phenomenological implications are well known and

studied in the literature. These operators preserve chiral symmetry, providing a structural

reason why the new strong interaction associated with compositeness does not percolate to

the SM fermion masses and Yukawas. The second scenario has instead a well-defined limit

where the dimension-6 effects vanish and compositeness is manifested through dimension-8

operators. One certain realization is associated with non-linearly realized supersymmetry;

that is, the fermions are composite pseudo-Goldstini.

In this article we built the EFT of N Goldstini and discussed how to embed the SM

fermions in such a framework. In particular, the SM gauge and flavor groups are subgroups

of the R-symmetry ⊂ U(N ). The SM gauge and Yukawa couplings explicitly break super-

symmetry, these breakings propagating to other observables – in particular they generate the

dimension-6 operators that were forbidden in the exact supersymmetric limit – but can be

treated as perturbative and we estimated their sizes.

We compared this goldstino-compositeness with the standard chiral-compositeness, con-

fronting them both with experimental data in the form of measurements of dijet distributions

at the LHC or e+e− scattering at LEP (in addition to other low-energy observables). Wilson

coefficients of the standard four-fermion operators scale as(g(6)∗ /m

(6)∗)2

, while the dimension-8

as (g(8)∗ )2/(m

(8)∗ )4, in terms of the physical scales and couplings of the new dynamics. Then,

an experiment performing at energy Eexp provides constraints that naively relate as,

m(8)∗ ∼ m(6)

∗

(Eexp

m(6)∗

)1/2(g(8)∗

g(6)∗

)1/2

, (71)

an intuition that we confirmed with dedicated analyses. Since Eexp/m(6)∗ � 1 in a sensible

EFT, for g(8)∗ ∼ g

(6)∗ bounds on goldstino-compositeness are always poorer than those on chiral-

compositeness: by a factor of 4 for light quarks (tested at LHC) and by a factor of 20 for light

leptons (tested at LEP-2). For pseudo-Goldstini, the onset of the new physics interactions is

so dramatic that energy, rather than accuracy, plays the crucial role in experimental searches.

As a result, electrons could be pseudo-Goldstini already at scales & 2 TeV, while light quarks

require scales & 10 TeV – from a phenomenological point of view goldstino-compositeness is

therefore on a better footing than chiral-compositeness, which requires scales & 40 TeV.

The scenarios with SM fermions as pseudo-Goldstini bring new interesting questions for

future research. From an experimental point of view, these scenarios provide novel and

alternative benchmarks to compare the performance of future colliders (such as FCCee, ILC

or CLIC in e+e− scattering), in which the importance of energy over accuracy is additionally

emphasized. Moreover, if other SM species are also composite, novel effects can be expected

in neutral diboson pair production, which we discussed in section 5.4. These probes are very

clean experimentally but they have always suffered from the lack of concrete BSM scenarios

31

that would make them interesting: goldstino-compositeness provides a plethora of new effects

that can be searched for in this type of processes (effects that, even if of the same EFT order

as the ones studied so far, are complementary, better measurable and better motivated).

From a theoretical point of view goldstini-compositeness relies on the existence of approxi-

mate and emergent supersymmetries. While proof-of-principle examples of this possibly exist,

it would be interesting to set this on a firmer ground. In this roadmap, the incorporation of a

solution to the hierarchy problem through extended suspersymmetry would be an important

additional target.

Acknowledgments

We are happy to acknowledge important conversations with Zohar Komargodski, Alexander

Monin, Alex Pomarol, Riccardo Rattazzi and Riccardo Torre. We thank Alexander Monin

for collaboration in the early stages of this work and Simone Alioli for discussions and for

providing us help with POWHEG. B.B. thanks Marco Cirelli and the LPTHE for the kind

hospitality during the completion of this work. B.B. is supported in part by the MIUR-FIRB

grant RBFR12H1MW “A New Strong Force, the origin of masses and the LHC”.

Dileptons at LEP

The set of dimension-8 operators from goldstino-compositeness of leptons relevant at LEP is

Oee = (∂ν eRγµeR)(eRγµ∂

νeR) O`ee ⊃ −(∂ν eLγµeL)(∂ν eRγµeR) + h.c.

Oeψ = (∂ν eRγµeR)(ψRγµ∂

νψR) + h.c. O`eψ ⊃ −(∂ν eLγµeL)(∂νψRγµψR) + h.c.

O`e`e ⊃ (∂ν eLγµeL)(eLγµ∂

νeL) O`ψe ⊃ −(∂νψLγµψL)(∂ν eRγµeR) + h.c.

O`e`ψ ⊃ (∂ν eLγµeL)(ψLγµ∂

νψL) + h.c. , (72)

where ψ = µ, τ and in each particular scenario cij = 1/2F 2. Requiring maximal R-symmetry

implies in particular cee = ceµ = ceτ , c`e`e = c`e`µ = c`e`τ and c`ee = c`eµ = c`µe = c`eτ = c`τ e.

These operators contribute to the differential dilepton cross sections,

dσ

d cos θ(e+e− → e+e−)BSM =

u

8πs

[(t2

s+s2

t

)e2 +

(t2

s−m2Z

+s2

t−m2Z

)gZeRg

ZeL

]c`ee

u3

16πs

[(1

s+

1

t

)e2(cee + c`e`e) +

(1

s−m2Z

+1

t−m2Z

)((gZeR)2cee + (gZeL)2c`e`e

)]+

u2

32πs

[u2(c2ee + c2`e`e) + 2(t2 + s2)c2`ee

], (73)

dσ

d cos θ(e+e− → ψ+ψ−)BSM = − u2t

16πs

[1

se2(ceψ + c`e`ψ) +

1

s−m2Z

((gZeR)2ceψ + (gZeL)2c`e`ψ

)]+

u2t

16πs

[1

se2 +

1

s−m2Z

gZeRgZeL

](c`eψ + c`ψe)+

u2t2

32πs(c2eψ + c2`e`ψ + c2`eψ + c2`ψe) , ψ = µ, τ

32

where s, t, u are the Mandelstam variables (s+ t+u = 0 in the massless approximation) and

t = −s(1− cos θ)/2 and u = −s(1 + cos θ)/2, with θ is the CM scattering angle.

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